Biomedical Engineering Reference
In-Depth Information
7.1.3 Listening to Numerical Solutions
With the methods described above, we can compute a good approximation
to the dynamics of the labial motion. As we have already discussed, the
pressure fluctuations at the input of the trachea
p
i
(
t
) are the result of two
perturbations: those induced by the modulation of the airflow due to labial
motion
s
(
x
(
t
)
,y
(
t
)), and the pressure waves that return to the base of the
trachea after being partially reflected back at the beak,
p
back
:
p
i
(
t
)=
s
(
x
(
t
)
,y
(
t
)) +
p
back
(
t
−
L/c
)
,
(7.14)
p
back
(
t
)=
−
γp
i
(
t
−
L/c
)
,
(7.15)
where
γ
is the coe
cient for internal reflection at the beak, and is equal to 1
in the ideal case of perfect reflection.
Computing
x
and
y
numerically allows us to obtain the air velocity at the
open end of the tract. In order to do so, we remember that
ρ
0
∂
v
/∂t
,
and use this relation to compute the velocity. The total pressure at a small
distance
dx
from the end of the tube is
∇
p
=
−
p
tot
(
x
=
L
−
dx
)=
p
back
(
t
−
dt
)+
p
i
(
t
−
(
T
−
dt
))
,
(7.16)
where
dx
=
cdt
. Since, at the boundary,
p
back
(
t
)=
−
p
i
(
t
−
T
), we can write
dt
)) = 2
dp
i
p
tot
(
x
=
L
−
dx
)=
−
p
i
(
t
−
T
−
dt
)+
p
i
(
t
−
(
T
−
dt
|
t−T
dt .
(7.17)
Computing the gradient from
p
i
=
0
−
2(
dp
i
/dt
)
|
t−T
dt
∇
(7.18)
dx
and performing a temporal integration, we obtain the following for the ve-
locity at the open end of the tube:
2
cρ
0
v
=
p
i
(
t − T
)
.
(7.19)
We can use the numerically computed time series representing the air mo-
tion to excite a mechanical membrane and generate a physical sound wave.
Software players actually allow you to do this: they convert numerical instruc-
tions into electrical signals at the output of a sound card, which are capable
of driving a loudspeaker. The tricky part of this procedure is to discover the
format of the file that is interpreted by the player. One of these formats is
called “wav”. In this format, the sound is represented by a string of integers
in the interval (
32768
,
32767), in a binary format (i.e., each value of
p
is
represented by the coe
cients
b
i
,where
p
=
b
i
2
i
). In this way, we can do
the following.
−
1. Integrate the equations of the labial motion
x
(
t
).
